2013年2月26日火曜日

【メモ】 Programming the Universe: A Quantum Computer Scientist Takes on the Cosmos (Seth Lloyd)

グッとくる文章だけ引用。

The quantum-computational nature of the universe dictates that the details of the future are intrinsically unpredictable. They can be computed only by a computer the size of the universe itself. Otherwise, the only way to discover the future is to wait and see what happens. (p.4)

After the first controlled-NOT, the demon's bit is perfectly correlated with the state of the gas bit. That is, the demon now "knows" the value of the gas bit. More precisely, as far as the demon is concerned, the entropy of the gas bit is zero, because it is conditioned on the state of the demon's bit after the operation, a state the demon knows. (pp.95-96)

マックスウェルのデーモンの思考実験では、デーモンを物理的なシステムとすると、必ず気体の系と相互作用しなければならなず、相互作用の過程(ここではcontrolled-NOTの演算)の前後を通して、気体粒子に関する情報が必然的に(デーモンの中で)保存される、ということ。だからデーモンを含めた系全体として見れば、エントロピーは減少していない。

Even if the underlying laws of physics were fully deterministic, however, the computational ability of simple systems such as colliding spheres implies that to perform the type of simulation Laplace envisaged, the calculating demon would have to have at least as much computational power as the universe as a whole. Since, as we shall see, computational power requires physical resources, Laplace's demon would have to use at least as much space, time, and energy as the universe itself. (p.98)

The reason [that bigger things tend to behave more "classically" and less quantum-mechanically] lies not so much with the physical size of the objects as with its visibility. The bigger something is, the more interactions it tends to have with its surroundings, thus the easier it is to detect. In order to go through both slits at once and produce the interference pattern, a particle must pass through the slits undetected. (p.107)

All that is required to destroy the interference pattern is for some system, no matter how small, to get information about the position of the particle. (...) It is now clear why big things tend to show up in one place or another, but not both. Pebbles, people, and planets are constantly interacting with their surroundings. Each interaction with an electron, a molecule of air, a particle of light tends to localize a system. Big things interact with lots of little things, each of which gets information about the location of the big thing. As a result, big things tend to appear here or there instead of here and there at the same time. (p.108)

The process by which the environment destroys the wavelike nature of things by getting information about a quantum system is called "decoherence." (p.108)

The information about which slit the particle went through infects first the detector, then me. If I write you a letter telling you whether or not I heard a click, then when you receive the letter your relative state will reflect what happened: |Seth wrote me to say he heard a click> or |Seth wrote me to say he didn't hear a click>. Now you have become entangled with the particle, the detector, and me. After the measurement, the information about its outcome spreads to infect whatever it comes into contact with. (p.122)

because the behavior of elementary particles can be mapped directly onto the behavior of qubits interacting via logic operations, a simulation of the universe on a quantum computer is indistinguishable from the universe itself. (p.154)

To look at this from another angle, numbers produced by short programs are more likely to appear as outputs of the [programming] monkey's computer than numbers that can be produced only by long programs. Many beautiful and intricate mathematical patterns - regular geometric shapes, fractal patterns, the laws of quantum mechanics, elementary particles, the laws of chemistry - can be produced by short computer programs. (...) Algorithmically probable things are just those things that exhibit large amounts of regularity, structure, and order. (p.184)

この説明が実際の宇宙で意味を成すためには、コンピュータと猿のそれぞれの対応物が必要だが、Lloydによると、コンピュータは量子力学の法則、猿は量子ゆらぎに対応しているらしい。ということは、宇宙の発生(計算の始まり)に先立って存在していなければならないのは、量子力学の法則(あるいは重力を織り込んだ量子重力理論?)に限る、ということだろう。しかし上の引用では、量子力学の法則も、ランダムなゆらぎの結果生じるかのように書いてある。何が計算に先立っていて、何が計算の結果生じるのか、はっきりしてよLlyodさん!

2013年2月1日金曜日

【メモ】 Thomas Nagel, The Last Word

[Chapter 2 - Why We Can't Understand Thought From the Outside]

However, the pursuit of self-awareness breaks down if we try to extend this kind of external psychological criticism of ourselves to the limit — which must happen if we entertain the possibility that nothing in human thought really qualifies as reason in the strong sense I wish to defend. For we are then supposed to consider the completely general possibility that there are contingent and local explanations of the sources of all our convictions, explanations that do not provide justifications as strong as reason would provide, if there were such a thing. And the question is, what kind of thought is this? It purports to be a view of ourselves from the outside, as creatures subject to various psychological influences and prey to certain habits, but what are we supposed to be relying on in ourselves to form that view? (p.14)

There are some types of thoughts that we cannot avoid simply having — that it is strictly impossible to consider merely from the outside, because they enter inevitably and directly into any process of considering ourselves from the outside, allowing us to construct the conception of a world in which, as a matter of objective fact, we and our subjective impressions are contained. (p.20)

It is hard to be satisfied with giving the last word to certain ordinary statements or forms of reasoning. If one rejects all relativizing qualifications, it is terribly tempting to add something else in their place: "2 + 2 = 4 and cruelty is evil, not just for us, but absolutely." But if this attempts to go beyond the denial of the qualification, it may, in the immortal words of Bernard Williams, be one thought too many — with the unfortunate implication that unless something positive can be put in that space, we will be left with subjectivism after all. It would be better if we could just come to a stop with certain kinds of judgments and arguments, which neither admit nor require further qualification. But that seems to demand a level of philosophical will‐power that is beyond most of us. (p.35)

[Chapter 3 - Language] 

Perhaps it is possible to understand the statement "I obey the rule blindly" in this way: It might be said that if I think that what I'm doing is just something I can't help, I am not really obeying the rule of addition blindly. To obey it blindly could be taken to mean simply drawing the conclusion which it mandates, with no further explanation than that that is the right answer. (p.49)

"We think we must find some facts, the recognition of which would not require that we already speak and understand a language, and some rules which would tell us what, given those facts, it was correct to say. Familiar, everyday statements of what a particular expression means cannot serve. They make essential use of words that are already "alive", that already have a meaning, so they seem incapable of explaining in the right way how any words come to have any meaning or come to be understood at all." (Barry Stroud, "Wittgenstein on Meaning, Understanding, and Community," in R. Haller and J. Brandl eds. Wittgenstein - Towards a Re‐Evaluation: Proceedings of the 14th International Wittgenstein Symposium (Hölder‐Pichler‐Tempsky, 1990), p.35.

[Chapter 4 - Logic]

To say that we cannot get outside them [simple logical thoughts] means that the last word, with respect to such beliefs, belongs to the content of the thought itself rather than to anything that can be said about it. No further comments on its origin or psychological character can in any way qualify it, in particular not the comment that it is just something I cannot help believing, or that it occupies a hierarchically dominant position in my system of beliefs. All that is secondary to the judgment itself. (pp.64-65)

Thought itself has priority over its description, because its description necessarily involves thought. The use of language has priority over its analysis, because the analysis of language necessarily involves its use. And in general, every external view of ourselves, every understanding of the contingency of our makeup and our responses as creatures in the world, has to be rooted in immediate first‐order thought about the world. However successfully we may get outside of ourselves in certain respects, thereby subjecting ourselves to doubt, criticism, and revision, all of it must be done by some part of us that we haven't got outside of, which simply has the thoughts, draws the inferences, forms the beliefs, makes the statements. (pp.65-66)

When I try to regard such a thought as a mere phenomenon, I cannot avoid also thinking its content—cannot retreat to thinking of it merely as words or pictures going through my head, for example. (p.66)

The subjectivist would no doubt reply that he can avoid offending against common sense, since he is merely analyzing what we ordinarily say, not recommending that we change it. (...) But this reply is useless. The reason it will not work is that the subjectivist always has something further to say, which does not fit into this framework but is supposed to be a comment on the significance and ultimate basis (in human practices) of the whole thing. And that comment simultaneously contradicts the true content of the original statements of reason, and contradicts itself by being intelligible only as an objective claim not grounded merely in our inescapable responses. (pp.66-67)

There is a general moral to be drawn from these observations, a moral that applies also to forms of reasoning very different from the simple, selfevident principles we have been considering so far, and it is this: Reflection about anything leads us inexorably to certain thoughts in which "I" plays no part—thoughts that are completely free of first‐person content. (This can be understood to include the first person plural for good measure.) Such "impersonal" thoughts are simply misrepresented by any attempt to say that the real ground of their truth or necessity is that we can't help having them, or that this is one of our fundamental and not further grounded responses or practices — to reinterpret or diagnose them in a personal or communal form. (p.67)

While they [free-standing thoughts, including those of logic and arithmetic] are had by us, they do not in any way refer to us, even implicitly. It is in this region of impersonal thoughts that do not depend on any personal ones that the operation of reason must be located. Reason, so understood, permits us to develop the conception of the world in which we, our impressions, and our practices are contained, because it does not depend on our personal perspective. We cannot judge any type of thought to be merely personal except from a standpoint that is impersonal. The aim of situating everything in a non‐firstperson framework — a conception of how things are — is one to which there is no alternative. (p.68)

This is the heart of the issue over the scope of reason, which includes those general forms or methods of impersonal thought, whatever they are, that we reach at the end of every line of questioning and every search for justification, and that we cannot in the end consider merely as a very deeply entrenched aspect of our point of view. I have been discussing particular logical and arithmetical examples, but the real character of reason is not found in belief in a set of "foundational" propositions, nor even in a set of procedures or rules for drawing inferences, but rather in any forms of thought to which there is no alternative. (pp.68-69)

This thought [that the numbers we use to count things in everyday life are merely the first part of a series that never ends] is a paradigm of the way reason allows us to reach vastly beyond ourselves. The local, finite practice of counting contains within itself the implication that the series is not completable by us: It has, so to speak, a built‐in immunity to attempts at reduction. Though our direct acquaintance with and designation of specific numbers is extremely limited, we cannot make sense of it except by putting them, and ourselves, in the context of something larger, something whose existence is independent of our fragmentary experience of it. Yet we draw this access to infinity out of our distinctly finite ability to count, in virtue of its evident incompleteness. When we think about the finite activity of counting, we come to realize that it can only be understood as part of something infinite. The idea of reducing the apparently infinite to the finite is therefore ruled out: Instead, the apparently finite must be explained in terms of the infinite. The reason this is a model for the irreducibility of reason in general is that it illustrates the way in which the application of certain concepts from inside overpowers the attempt to grasp that application from outside and to describe it as a finite and local practice. It may look small and "natural" from outside, but once one gets inside it, it opens out to burst the boundaries of that external naturalistic view. It is like stepping into what looks like a small windowless hut and finding oneself suddenly in the middle of a vast landscape stretching endlessly out to the horizon. And it is precisely by posing the reductive question that we come to see this. We discover infinity when we ask whether these numbers we can name are all there is, whether we can understand counting as just a finite human practice in which speakers of the language come to relatively easy agreement. From inside the practice itself comes a negative answer: The view from inside dominates the view from outside, unless the latter somehow expands to include a version of the former. (There is an analogy here with the philosophy of mind: An external view of the mental cannot be adequate unless it expands to incorporate in some form the internal view.) (pp.71-72)

Perhaps there is something wrong with the hope of arriving at a complete understanding of the world that includes an understanding of ourselves as beings within it possessing the capacity for that very understanding. I think something of the kind must be true. There are inevitably going to be limits on the closure achievable by turning our procedures of understanding on themselves. If that is so, then the outer boundaries of our understanding will always be reached in unqualified, objective reasoning about the real world rather than in the interpretation and expression of our own perspective — personal or social. To engage in such reasoning is to try to bring one's individual thoughts under the control of a universal standard that prescribes to each person those beliefs, available from his point of view, which can form part of a consistent set of objective beliefs dispersed over all rational persons. It enables us all to live in part of the truth. (p.76)

[Chapter 5 - Science] 

A thoroughly realistic conception of natural laws will have to try to interpret them also in ways that are independent of any particular point of view or observational standpoint — otherwise they might be merely ways of systematizing our observations. To be truly mind‐independent, the laws — and not just the events they govern — must be perspective‐free and must explain why things appear as they do from different vantage points within the world. In modern physics, this idea constrains the development of theories through a requirement of symmetry — that the real natural order should be identified with what is invariant from the points of view of all observers, so that whatever their situation, they can all arrive at the same description of the common reality in which they are situated. The requirement applies not just to particular states of affairs but to general laws. It was this demand for symmetry or invariance in the description of nature that led Einstein first to the special and then to the general theory of relativity and that has apparently had a major role in shaping quantum theory. (p.83)

This position [Putnam's internal realism] adds a qualification to our empirical claims that I believe is inconsistent with their content, in the same way that subjectivism about logic is inconsistent with its content. Furthermore, the only way to make literal sense of the qualification is in terms of a conception of the world and our place in it which is not itself subjective but according to which our entire system of substantive beliefs, by contrast, is. If we wish to adopt a view of the world that places our own thoughts within it and also answers to the demand for a natural order, it will have to be a view without such qualifications, subject to the same kind of reasoning about how things are that applies elsewhere — not a merely "internal" view. (p.88)

The belief that the world is orderly, and that our sense of what constitutes order (what properties are usable in the formulation of laws and inferences) is an indication of how the world is organized, is well confirmed in some areas, where we have discovered that the hypotheses to which we are led — theories about unobservables and the laws governing them — predict observations that are not themselves explainable by our belief in those hypotheses. The fact that observation is "theory‐laden" seems to me an insignificant point which in no way tends to show that the process of confirming theories by observation is circular or nonobjective. It may require some theory, of telescopes or of photography, to interpret the astronomical photographs that show the bending of light rays by the sun's gravitational field, but the crucial observation — that the images of the stars near the sun are displaced outward — is not dependent on the theory which it confirms — namely, the general theory of relativity. The possibility of noncircular confirmation is also, I think, the answer to doubts about the role of our natural sense of similarity in determining what counts for us as a regularity or law. The fact is that we can demote a similarity or a kind to the status of mere appearance, or similarity for us, only if it is shown to be not systematically connected with other observed regularities. But if some of the regularities we observe, including those revealed by measurement, turn out to be systematically correlated with others that emerge from different types of observation or measurement, then the most plausible hypothesis is that these are not artefacts of our perspective on the world but, rather, products of the world's systematic interaction with us. (pp.89-90)

Yet it has to be granted that the empirical confirmation of the supposition that the world is orderly and that particular phenomena can be explained by general laws has something inevitably circular about it. (...) But there is really no alternative. The attempt to reconstrue the ordered world picture as a projection of our minds founders on the need to place ourselves in the world so ordered. In trying to make sense of this relation, we are inevitably led to employ the same kind of reasoning, based on the search for order. Even if we decide that some of our apprehensions of order are illusions or errors, that will be because a better theory, by the same standards, can explain them away. Ultimately, all we can do is think about how the world is, including ourselves and our relation to the rest of it; and the only way to do that is to place our own experience in a larger setting that is suggested by the usual sort of empirical reasoning. It is certainly not a necessary truth that the world is orderly, let alone that we can understand its order. Substantial aspects of reality may never submit to this kind of intellectual grasp. But anything we can know about must be at least related in an orderly way to us, and an amazing amount has proved to be within our reach; given our achievements so far, it is reasonable to try to continue. (pp.91-92)

Kantian transcendental idealism is a thesis not about the phenomenal world but about the relation of the phenomenal world to the world as it is in itself. But since it says that ordinary scientific reasoning applies only to the phenomenal world, it exempts itself from the usual conditions of assessment. The thesis of transcendental idealism is not itself one of the synthetic a priori judgments whose validity it purports to explain, but it is an a priori claim all the same, based on the conviction that there is no other way things could be — that it is inconceivable that we should be able to use empirical evidence to find out about things as they are in themselves. (p.93)

I don't see how this proposal can be understood in a way that does not put it into competition with more mundane views about our place in the world and our relations with the rest of it, views that are supported by the ordinary methods of rational assessment and explanation. (p.94)

To the proposal that the order we appear to discover is just a framework we impose on experience, the inevitable, unexciting reply is that that does not seem a particularly likely explanation of the observed facts — that a more plausible account is that, to a considerable extent, the order that we find in our experience is the product of an order that is there independent of our minds. Applied to any real aspect of the natural order, the Kantian interpretation seems bizarre. (p.95)

[Chapter 7 - Evolutionary Naturalism and the Fear of Religion]

Charles Sanders Peirce is usually described as the founder of pragmatism, but that is no more accurate than describing Wittgenstein as the founder of logical positivism. (p.127)

"The only end of science, as such, is to learn the lesson that the universe has to teach it. In Induction it simply surrenders itself to the force of facts. But it finds . . . that this is not enough. It is driven in desperation to call upon its inward sympathy with nature, its instinct for aid, just as we find Galileo at the dawn of modern science making his appeal to il lume naturale . . . The value of Facts to it, lies only in this, that they belong to Nature; and nature is something great, and beautiful, and sacred, and eternal, and real, — the object of its worship and its aspiration." (C. S. Peirce, Reasoning and the Logic of Things, edited by Kenneth Laine Ketner, with introduction and commentary by Ketner and Hilary Putnam (Harvard University Press, 1992), pp.176-177)

Now I find these declarations not only eloquent but entirely congenial; but they have a radically antireductionist and realist tendency quite out of keeping with present fashion. And they are alarmingly Platonist in that they maintain that the project of pure inquiry is sustained by our "inward sympathy" with nature, on which we draw in forming hypotheses that can then be tested against the facts. Something similar must be true of reason itself, which according to Peirce has nothing to do with "how we think." If we can reason, it is because our thoughts can obey the order of the logical relations among propositions — so here again we depend on a Platonic harmony. The reason I call this view alarming is that it is hard to know what world picture to associate it with, and difficult to avoid the suspicion that the picture will be religious, or quasi‐religious. Rationalism has always had a more religious flavor than empiricism. Even without God, the idea of a natural sympathy between the deepest truths of nature and the deepest layers of the human mind, which can be exploited to allow gradual development of a truer and truer conception of reality, makes us more at home in the universe than is secularly comfortable. The thought that the relation between mind and the world is something fundamental makes many people in this day and age nervous. I believe this is one manifestation of a fear of religion which has large and often pernicious consequences for modern intellectual life. (pp.129-130)

But is the [evolutionary] hypothesis really compatible with continued confidence in reason as a source of knowledge about the nonapparent character of the world? In itself, I believe an evolutionary story tells against such confidence. Without something more, the idea that our rational capacity was the product of natural selection would render reasoning far less trustworthy than Nozick suggests, beyond its original "coping" functions. There would be no reason to trust its results in mathematics and science, for example. (...) Unless it is coupled with an independent basis for confidence in reason, the evolutionary hypothesis is threatening rather than reassuring. It is consistent with continued confidence only if it amounts to the hypothesis that evolution has led to the existence of creatures, namely us, with a capacity for reasoning in whose validity we can have much stronger confidence than would be warranted merely from its having come into existence in that way. I have to be able to believe that the evolutionary explanation is consistent with the proposition that I follow the rules of logic because they are correct — not merely because I am biologically programmed to do so. But to believe that, I have to be justified independently in believing that they are correct. And this I cannot be merely on the basis of my contingent psychological disposition, together with the hypothesis that it is the product of natural selection. I can have no justification for trusting a reasoning capacity I have as a consequence of natural selection, unless I am justified in trusting it simply in itself — that is, believing what it tells me, in virtue of the content of the arguments it delivers. If reason is in this way self‐justifying, then it is open to us also to speculate that natural selection played a role in the evolution and survival of a species that is capable of understanding and engaging in it. But the recognition of logical arguments as independently valid is a precondition of the acceptability of an evolutionary story about the source of that recognition. This means that the evolutionary hypothesis is acceptable only if reason does not need its support. At most it may show why the existence of reason need not be biologically mysterious.(pp.135-136)

 So my conclusion about an evolutionary explanation of rationality is that it is necessarily incomplete. Even if one believes it, one has to believe in the independent validity of the reasoning that is the result. (p.137)